Applications of Alternating Direction Solver for simulations of time-dependent problems

Authors

  • Marcin Mateusz Łoś AGH University of Science and Technology
  • Maciej Paszyński AGH University of Science and Technology

DOI:

https://doi.org/10.7494/csci.2017.18.2.117

Keywords:

FEM, Isogeometric Analysis, Alternating Direction Solver

Abstract

This paper deals with application of Alternating Direction solver (ADS) to nonstationary
linear elasticity problem solved with isogeometric FEM. Employing
tensor product B-spline basis in isogeometric analysis under some restrictions
leads to system of linear equations with matrix possessing tensor product structure.
Alternating Direction Implicit algorithm is a direct method that exploits
this structure to solve the system in O (N ), where N is a number of degrees
of freedom (basis functions). This is asymptotically faster than state-of-theart
general purpose multi-frontal direct solvers. In this paper we also present
the complexity analysis of ADS incorporating dependence on order of B-spline
basis.

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Author Biographies

Marcin Mateusz Łoś, AGH University of Science and Technology

PhD student of Informatics, Faculty of Computer Science, Electronics and Telecommunications at AGH University of Science and Technology

Maciej Paszyński, AGH University of Science and Technology

Professor at Faculty of Computer Science, Electronics and Telecommunications at AGH University of Science and Technology

References

Anderson E., Bai Z., Dongarra J., Greenbaum A., McKenney A., Croz J.D., Hammerling S., Demmel J., Bischof C., Sorensen D.: LAPACK: A Portable Linear Algebra Library for High-performance Computers. In: Proceedings of the 1990 ACM/IEEE Conference on Supercomputing, Supercomputing ’90, pp. 2–11. IEEE Computer Society Press, Los Alamitos, CA, USA, 1990. ISBN 0-89791412-0. URL http://dl.acm.org/citation.cfm?id=110382.110385.

Birkhoff G., Varga R.S., Young D.: Alternating Direction Implicit Methods. Advances in Computers, vol. 3, pp. 189 – 273. Elsevier, 1962.

Collier N., Pardo D., Dalcin L., Paszynski M., Calo V.: The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers. In: Computer Methods in Applied Mechanics and Engineering, vol. 213216(0), pp. 353 – 361, 2012. ISSN 0045-7825.

Cottrell J.A., Hughes T.J.R., Bazilevs Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley Publishing, 1st ed., 2009. ISBN 0470748737, 9780470748732.

Duff I.S., Reid J.K.: The Multifrontal Solution of Indefinite Sparse Symmetric Linear. In: ACM Trans. Math. Softw., vol. 9(3), pp. 302–325, 1983. ISSN 00983500.

D.W. Peaceman H.R.J.: The numerical solution of parabolic and elliptic differential equations. In: Journal of Society of Industrial and Applied Mathematics, (3), 1955.

E.L. Wachspress G.H.: An alternating-direction-implicit iteration technique. In: Journal of Society of Industrial and Applied Mathematics, (8), 1960.

Gao L., Calo V.M.: Fast isogeometric solvers for explicit dynamics. In: Computer Methods in Applied Mechanics and Engineering, vol. 274(0), pp. 19 – 41, 2014. ISSN 0045-7825.

Gao L., Calo V.M.: Preconditioners based on the Alternating-Direction-Implicit algorithm for the 2D steady-state diffusion equation with orthotropic heterogeneous coefficients. In: Journal of Computational and Applied Mathematics, vol. 273(0), pp. 274 – 295, 2015. ISSN 0377-0427.

Golub G.G., Loan C.F.V.: Matrix Computations. The Johns Hopkins University Press, 4th ed., 2013.

Hawkins-Daarud A., Prudhomme S., van der Zee K., Oden J.: Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth. In: Journal of Mathematical Biology, vol. 67(6-7), pp. 1457–1485, 2013. ISSN 0303-6812. URL http://dx.doi.org/10.1007/ s00285-012-0595-9.

Hawkins-Daarud A., van der Zee K.G., Tinsley Oden J.: Numerical simulation of a thermodynamically consistent four-species tumor growth model. In: International Journal for Numerical Methods in Biomedical Engineering, vol. 28(1), pp. 3–24, 2012. ISSN 2040-7947. URL http://dx.doi.org/10.1002/cnm.1467.

Hughes T., Cottrell J., Bazilevs Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. In: Computer Methods in Applied Mechanics and Engineering, vol. 194(3941), pp. 4135 – 4195, 2005. ISSN 00457825.

J. Douglas H.R.: On the numerical solution of heat conduction problems in two and three space variables. In: Transactions of American Mathematical Society, (82), 1956.

Los M., Wozniak M., Paszynski M., Dalcin L., Calo V.M.: Parallel alternating direction preconditioner for isogeometric simulations of explicit dynamics. 1st Pan-American Congress on Computational Mechanics, Buenos Aires, April 27– 29.

Los M., Wozniak M., Paszynski M., Dalcin L., Calo V.M.: Dynamics with matrices possesing Kronecker product structure. In: Procedia Computer Science, (51), pp. 286 – 295, 2015.

Marsh D.: Applied Geometry for Computer Graphics and CAD. Springer Undergraduate Mathematics Series. Sprlinger-Verlag London, 2005.

N. Collier L. Dalcin V.C.: PetIGA: High-Performance Isogeometric Analysis. In: arxiv, (1305.4452), 2013. Http://arxiv.org/abs/1305.4452.

Piegl L., Tiller W.: The NURBS Book (2nd ed.). Springer-Verlag New York, Inc., New York, NY, USA, 1997. ISBN 3-540-61545-8.

Wozniak M., Los M., Paszynski M., Dalcin L., Calo V.M.: Parallel fast isogeometric solvers for explicit dynamic. In: Computing and Informatics, 2015.

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Published

2017-06-23

How to Cite

Łoś, M. M., & Paszyński, M. (2017). Applications of Alternating Direction Solver for simulations of time-dependent problems. Computer Science, 18(2), 117. https://doi.org/10.7494/csci.2017.18.2.117

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